lim da raíz cúbica de (t+a)²-raíz cúbica de a² /t t tende a 0

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Para encontrarmos o limite da função, devemos realizar os cálculos abaixo:

\(\begin{align} & f=\sqrt[3]{(t+{{a}^{2}})}-\sqrt[3]{\frac{{{a}^{2}}}{t}} \\ & \underset{t\to 0}{\mathop{\lim }}\,f=\sqrt[3]{(t+{{a}^{2}})}-\sqrt[3]{\frac{{{a}^{2}}}{t}} \\ & \underset{t\to 0}{\mathop{\lim }}\,f=\sqrt[3]{(\frac{t}{t}+{{\frac{a}{t}}^{2}})}-\sqrt[3]{\frac{{{\frac{a}{t}}^{2}}}{\frac{t}{t}}} \\ & \underset{t\to 0}{\mathop{\lim }}\,f=\sqrt[3]{(1+{{\frac{a}{0}}^{2}})}-\sqrt[3]{\frac{{{\frac{a}{0}}^{2}}}{1}} \\ & \underset{t\to 0}{\mathop{\lim }}\,f=\sqrt[3]{1}-0 \\ & \underset{t\to 0}{\mathop{\lim }}\,f=1 \\ \end{align}\ \)

Portanto, o limite da função será \(\boxed{f = 1}\).